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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "This worksheet illustrate
s the Cayley trick, showing the link between triangulations and mixed \+
subdivisions." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 
"with(plots): with(plottools):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
22 "1. The Cayley polytope" }}{PARA 0 "" 0 "" {TEXT -1 172 "The Cayley
 polytope of an r-tuple of polytopes is defined as the convex hull obt
ained by placing the polytopes in the tuple on the vertices of an (r-1
)-dimensional simplex." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "We illu
strate this construction on two polygons so that we can show the Cayle
y polytope as a 3-D object." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "radius := 0.03: thick := 3:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 49 "The vertices of the first polygon lie at level \+
0:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A1
v0 := [0,0,0]: p_A1v0 := sphere(A1v0,radius):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 108 "t2_A1v0 := textplot3d([A1v0[1],A1v0[2]-0.1,A1v0
[3],`(0,0)`],font=[TIMES,BOLD,12],color=black,align = BELOW):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "t3_A1v0 := textplot3d([A1v0
[1],A1v0[2],A1v0[3]+0.05,`(0,0,0)`],font=[TIMES,BOLD,12],color=black,a
lign = ABOVE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A1v1 := [
1,2,0]: p_A1v1 := sphere(A1v1,radius):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "t2_A1v1 := textplot3d([A1v1[1],A1v1[2]+0.1,A1v1[3],`
(1,2)`],font=[TIMES,BOLD,12],color=black,align = ABOVE):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "t3_A1v1 := textplot3d([A1v1[1],A1v
1[2],A1v1[3]-0.1,`(1,2,0)`],font=[TIMES,BOLD,12],color=black,align = B
ELOW):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A1v2 := [3,1,0]: \+
p_A1v2 := sphere(A1v2,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 108 "t2_A1v2 := textplot3d([A1v2[1]+0.1,A1v2[2],A1v1[3],`(3,1)`],f
ont=[TIMES,BOLD,12],color=black,align = RIGHT):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 110 "t3_A1v2 := textplot3d([A1v2[1],A1v2[2],A1v1[3
]-0.1,`(3,1,0)`],font=[TIMES,BOLD,12],color=black,align = BELOW):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "There are three edges:" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A1e01 := line(A1
v0,A1v1,color=green,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "A1e02 := line(A1v0,A1v2,color=green,thickness=thick):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A1e12 := line(A1v1,A1v2
,color=green,thickness=thick):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 
"The first polygon is defined by three vertices and three edges:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "polygon
1 := [p_A1v0,p_A1v1,p_A1v2,A1e01,A1e02,A1e12]:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "labels1_2d := t2_A1v0,t2_A1v1,t2_A1v2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "labels1_3d := t3_A1v0,t3_A1v
1,t3_A1v2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "display(polyg
on1,labels1_2d,orientation=[-90,0],scaling=constrained);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 35 "The second polygon lies at level 1:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A2v0 :=
 [0,0,1]: p_A2v0 := sphere(A2v0,radius):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "t2_A2v0 := textplot3d([A2v0[1],A2v0[2]-0.1,A2v0[3],`
(0,0)`],font=[TIMES,BOLD,12],color=black,align = BELOW):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "t3_A2v0 := textplot3d([A2v0[1],A2v
0[2],A2v0[3]+0.05,`(0,0,1)`],font=[TIMES,BOLD,12],color=black,align = \+
ABOVE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A2v1 := [1,1,1]:
 p_A2v1 := sphere(A2v1,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 108 "t2_A2v1 := textplot3d([A2v1[1],A2v1[2]+0.1,A2v1[3],`(1,1)`],f
ont=[TIMES,BOLD,12],color=black,align = ABOVE):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 111 "t3_A2v1 := textplot3d([A2v1[1],A2v1[2],A2v1[3
]+0.05,`(1,1,1)`],font=[TIMES,BOLD,12],color=black,align = ABOVE):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A2v2 := [4,0,1]: p_A2v2 := \+
sphere(A2v2,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "t2
_A2v2 := textplot3d([A2v2[1],A2v2[2]-0.1,A2v2[3],`(4,0)`],font=[TIMES,
BOLD,12],color=black,align = BELOW):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 111 "t3_A2v2 := textplot3d([A2v2[1],A2v2[2],A2v0[3]+0.05,
`(4,0,1)`],font=[TIMES,BOLD,12],color=black,align = ABOVE):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Also the second polytope has three
 edges:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "A2e01 := line(A2v0,A2v1,color=red,thickness=thick):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A2e02 := line(A2v0,A2v2,color=red,t
hickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A2e12 :
= line(A2v1,A2v2,color=red,thickness=thick):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 67 "The second polygon also consists of three vertices and \+
three edges:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "polygon2 := [p_A2v0,p_A2v1,p_A2v2,A2e01,A2e02,A2e12]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "labels2_2d := t2_A2v0,t2_A2v
1,t2_A2v2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "labels2_3d :=
 t3_A2v0,t3_A2v1,t3_A2v2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
69 "display(polygon2,labels2_2d,orientation=[-90,0],scaling=constraine
d);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "trans_poly2 := plott
ools[translate](display(polygon2,labels2_2d,scaling=constrained),5,0,0
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "display(polygon1,labe
ls1_2d,trans_poly2,orientation=[-90,0],scaling=constrained);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Here we see the two polygons, prop
erly embedded in three space:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 106 "display(polygon1,labels1_3d,polygon2,labels2
_3d,orientation=[42,73],scaling=unconstrained,projection=0.9);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The parallel side edges of the Cay
ley polytope are" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "spe0 := line(A1v0,A2v0,color=blue):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "spe1 := line(A1v1,A2v1,color=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "spe2 := line(A1v2,A2v2,color
=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sides1 := [spe0,
spe1,spe2]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The skew side edge
s of the Cayley polytope are" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "sse0 := line(A1v0,A2v2,color=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sse1 := line(A1v1,A2v0,color
=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sse2 := line(A1v
1,A2v2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "side
s2 := [sse0,sse1,sse2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "
polytope := polygon1,labels1_3d,polygon2,labels2_3d,sides1,sides2:" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Here is the polytope:" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(polytope
,orientation=[42,73],projection=0.9);" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 50 "2. Scaling the polygons inside the Cayley polytope" }}
{PARA 0 "" 0 "" {TEXT -1 317 "Simplices of any triangulation of the Ca
yley polytope correspond to cells of a mixed subdivision.  In this sec
tion we illustrate how the scaling of the original polygons happens in
side simplices of the Cayley polytope.  In particular, we take simplic
es spanned by one of the original polygons and one opposite vertex." }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The first cell is the first botto
m polygon scaled with a factor:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 11 "fac := 1/4:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "A1m0 := A1v0*fac + A2v2*(1-fac): p_A1m0 := sphere(A1m
0,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A1m1 := A1v1*
fac + A2v2*(1-fac): p_A1m1 := sphere(A1m1,radius):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 63 "A1m2 := A1v2*fac + A2v2*(1-fac): p_A1m2 := \+
sphere(A1m2,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A1m
idpts := [p_A1m0,p_A1m1,p_A1m2]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "A1m01 := line(A1m0,A1m1,color=green,thickness=thick):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A1m02 := line(A1m0,A1m2
,color=green,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "A1m12 := line(A1m1,A1m2,color=green,thickness=thick):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "cell1 := A1midpts,A1m01,A1m0
2,A1m12:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "The first unmixed cel
l appears halfway in a simplex of the triangulation of the Cayley poly
tope:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 
"simplex1_vertices := A1midpts,p_A2v2:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "simplex1_edges := spe2,sse0,sse2:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 60 "simplex1 := simplex1_vertices,simplex1_edge
s,cell1,polygon1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "displa
y(simplex1,labels1_3d,t3_A2v2,orientation=[42,73],projection=0.9);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The second cell is the second top
 polygon scaled with a factor:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 63 "A2m0 := A1v1*fac + A2v0*(1-fac): p_A2m0 := s
phere(A2m0,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A2m1
 := A1v1*fac + A2v1*(1-fac): p_A2m1 := sphere(A2m1,radius):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A2m2 := A1v1*fac + A2v2*(1-f
ac): p_A2m2 := sphere(A2m2,radius):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "A2midpts := [p_A2m0,p_A2m1,p_A2m2]:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 51 "A2m01 := line(A2m0,A2m1,color=red,thickne
ss=thick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A2m02 := line
(A2m0,A2m2,color=red,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "A2m12 := line(A2m1,A2m2,color=red,thickness=thick):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "cell2 := A2midpts,A2m01,A
2m02,A2m12:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The second unmixed
 cells appears halfway in a simplex of the Cayley polytope:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simplex
2 := p_A1v1,cell2,polygon2,sse1,sse2,spe1:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 72 "display(simplex2,labels2_3d,t3_A1v1,orientation=[42
,73],projection=0.9);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Here we \+
show those two simplices inside the Cayley polytope:" }{MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "scaletope := polytope,
cell1,cell2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(sca
letope,orientation=[42,73],projection=0.9);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 90 "In the middle of the Cayley polytope we now see the sca
led polygons, scaled with a factor." }{MPLTEXT 1 0 0 "" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 22 "3. A mixed subdivision" }}{PARA 0 "" 0 "
" {TEXT -1 87 "We show a mixed subdivision of the two polygons by slic
ing the Cayley polytope in half." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
100 "We define the half lines in the Cayley polytope, first all those \+
leading to the first unmixed cell: " }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "hAe0 := line(A1v0,A1m0,color=blue):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "hAe1 := line(A1v1,A1m1,
color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "hAe2 := lin
e(A1v2,A1m2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 
"hsides1 := [hAe0,hAe1,hAe2]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "
and all those leading to the second unmixed cell:" }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "hAe0 := line(A1v1,A2m0,co
lor=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "hAe1 := line(
A1v1,A2m1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "h
Ae2 := line(A1v1,A2m2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "hsides2 := [hAe0,hAe1,hAe2]:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "halftope := polygon1,hsides1,hsides2:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now, there is one mixed cell:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "m00 := \+
A1v0*fac+A2v0*(1-fac): p_m00 := sphere(m00,radius):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "hm00 := line(A1v0,m00,color=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "m00A2 := line(m00,A1m0,color
=red,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "m
00A1 := line(m00,A2m0,color=green,thickness=thick):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "cell3 := p_m00,hm00,m00A1,m00A2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subdivision := halftope,cell
1,cell2,cell3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display(s
ubdivision,labels1_3d,orientation=[42,73],projection=0.9);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 104 "The picture above shows the whole mixed \+
subdivision, obtained after slicing the Cayley polytope in half." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "mm00 :=
 line(m00,A2v0,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "mm10 := line(A2m0,A2v0,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "mm12 := line(A1m1,A2v2,color=blue):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "mm22 := line(A1m0,A2v2,color=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "mixed_edge := A2e02,p_A2v0,p
_A2v2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "mixed_cell := mix
ed_edge,mm00,mm10,mm12,mm22:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "T
he mixed cell occurs halfway in the third simplex of the triangulation
 of the Cayley polytope:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "simplex3_vertices := p_A1v0,p_A1v1,p_A1m0,p_A1m1,p
_A2m0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "simplex3_edges :=
 A1e01,sse1,sse0,sse2,A1m01,A2m02:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "simplex3 := simplex3_vertices,simplex3_edges,mixed_ce
ll,cell3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "lt3_A1v0 := t
extplot3d([A1v0[1],A1v0[2],A1v0[3]-0.1,`(0,0,0)`],font=[TIMES,BOLD,12]
,color=black,align = BELOW):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "simplex3_labels := lt3_A1v0,t3_A1v1,t3_A2v2,t3_A2v0:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "display(simplex3,simplex3_labels,or
ientation=[42,73],projection=0.9);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "topdivision := subdivision,mixed_cell:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "top_labels := labels1_3d,t3_A2v0,t3
_A2v2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display(topdivisi
on,top_labels,orientation=[42,73],projection=0.9);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 107 "Here we have the complete picture: triangulation
 of the Cayley polytope with therein the mixed subdivision:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fulltop
e := topdivision,scaletope:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "display(fulltope,orientation=[42,73],projection=0.9);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 123 "We now look at the mixed subdivision in \+
the plane, first by looking at the cross section of the Cayley polytop
e from above:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "display(subdivision,labels1_3d,orientation=[-90,0],sc
aling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Then we ma
ke the picture in the plane:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "v0 := [0,0,0]: dv0 := sphere(v0,radius):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "v1 := [4,0,0]: dv1 := sphere
(v1,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "v2 := [1,2,
0]: dv2 := sphere(v2,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "v3 := [5,2,0]: dv3 := sphere(v3,radius):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "v4 := [2,3,0]: dv4 := sphere(v4,radius):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "v5 := [7,1,0]: dv5 := sphere
(v5,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "vertices :=
 dv0,dv1,dv2,dv3,dv4,dv5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "e01 := line(v0,v1,color=red,thickness=thick):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 45 "e23 := line(v2,v3,color=red,thickness=thick)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "e24 := line(v2,v4,colo
r=red,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "
e34 := line(v3,v4,color=red,thickness=thick):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "e02 := line(v0,v2,color=green,thickness=thick):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "e13 := line(v1,v3,color
=green,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 
"e15 := line(v1,v5,color=green,thickness=thick):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 47 "e35 := line(v3,v5,color=green,thickness=thick
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "edges := e01,e23,e24,
e34,e02,e13,e15,e35:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "dis
play(vertices,edges,orientation=[270,0],axes=frame,scaling=constrained
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This subdivision is regular
, as last plot we show the lower hull:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "lv4 := [2,3,1]: ldv4 := sphere(lv4,
radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "lv5 := [7,1,1]:
 ldv5 := sphere(lv5,radius):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "lvertices := vertices,ldv4,ldv5:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "e24 := line(v2,v4,color=red,linestyle=3,thickness=thi
ck):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "e34 := line(v3,v4,c
olor=red,linestyle=3,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "e15 := line(v1,v5,color=green,linestyle=3,thickness=t
hick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "e35 := line(v3,v5
,color=green,linestyle=3,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "le24 := line(v2,lv4,color=red,thickness=thick):" }}
}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "le34 := line(v3,lv4
,color=red,thickness=thick):" }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 49 "le15 := line(v1,lv5,color=green,thickness=thick)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "le35 := line(v3,lv5,co
lor=green,thickness=thick):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "ledges := e01,e02,e23,e13,e24,e34,e15,e35,le24,le34,le15,le35:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "display(lvertices,ledges,sc
aling=constrained,orientation=[265,70],axes=frame);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "5 41 0 0" 0 }{VIEWOPTS 1 1 
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }